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Although the use of quadratically modified periodograms as spectral estimators for the Gaussian case is well established in the literature, the linearly modified form has grown in popular use even when the data involved can be reasonably presumed to be Gaussian. The mean and variance of the two forms of estimators are calculated as well as the equivalent duration of the data window. It is shown that variance is always greater or the equivalent duration of the sample data length is always less for the linear case when the results are normalized with respect to the effective bandwidth. In addition, a generalized form of the Hanning window is considered.